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Properties

Label 2-360-72.11-c1-0-11
Degree $2$
Conductor $360$
Sign $0.277 - 0.960i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 − 0.518i)2-s + (0.424 + 1.67i)3-s + (1.46 + 1.36i)4-s + (0.5 + 0.866i)5-s + (0.311 − 2.42i)6-s + (3.88 + 2.24i)7-s + (−1.21 − 2.55i)8-s + (−2.63 + 1.42i)9-s + (−0.208 − 1.39i)10-s + (1.05 + 0.606i)11-s + (−1.66 + 3.03i)12-s + (1.71 − 0.987i)13-s + (−3.94 − 4.96i)14-s + (−1.24 + 1.20i)15-s + (0.277 + 3.99i)16-s − 5.28i·17-s + ⋯
L(s)  = 1  + (−0.930 − 0.366i)2-s + (0.245 + 0.969i)3-s + (0.731 + 0.682i)4-s + (0.223 + 0.387i)5-s + (0.127 − 0.991i)6-s + (1.46 + 0.847i)7-s + (−0.430 − 0.902i)8-s + (−0.879 + 0.475i)9-s + (−0.0660 − 0.442i)10-s + (0.316 + 0.182i)11-s + (−0.482 + 0.876i)12-s + (0.474 − 0.273i)13-s + (−1.05 − 1.32i)14-s + (−0.320 + 0.311i)15-s + (0.0694 + 0.997i)16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893365 + 0.671578i\)
\(L(\frac12)\) \(\approx\) \(0.893365 + 0.671578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.518i)T \)
3 \( 1 + (-0.424 - 1.67i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-3.88 - 2.24i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.05 - 0.606i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.71 + 0.987i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.28iT - 17T^{2} \)
19 \( 1 + 4.86T + 19T^{2} \)
23 \( 1 + (-1.40 - 2.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.20 - 7.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.04 + 3.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.71iT - 37T^{2} \)
41 \( 1 + (5.30 - 3.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.17 + 2.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.97 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.739T + 53T^{2} \)
59 \( 1 + (-1.70 + 0.986i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.81 + 2.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.58 - 2.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.40T + 71T^{2} \)
73 \( 1 + 6.69T + 73T^{2} \)
79 \( 1 + (11.2 + 6.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.00 + 2.89i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.71iT - 89T^{2} \)
97 \( 1 + (-8.61 + 14.9i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.38951663049400169951884304755, −10.69492350984870636398975238778, −9.781181247666045370795156384352, −8.820048580767687392032874364124, −8.364448312603128078043607697889, −7.18312445576856346281795031590, −5.72997878819973319953265898821, −4.58252844190642130171570036830, −3.11782653232544794653478932161, −1.95463388761218005444419556769, 1.12170193071329263348983341175, 2.07147903131750682298923262730, 4.27511078977925440618537039898, 5.80677298819836623539706077530, 6.65151869457183140107474226688, 7.74906366762380260080959638523, 8.329470550474897163542364552282, 8.987883756727089385046397406412, 10.44763085582450831039009803515, 11.09910654954566176070766645235

Graph of the $Z$-function along the critical line